Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in ${\bf R}\sp n$. II

We prove a uniqueness result for the positive solution of $\Delta u + f(u) = 0$ in ${\mathbb{R}^n}$ which goes to 0 at $\infty$. The result applies to a wide class of nonlinear functions $f$, including the important model case $f(u) = - u + {u^p}$ , $1 < p < (n + 2)/(n - 2)$. The result is proved by reducing to an initial-boundary problem for the ${\text{ODE}}\;u'' + (n - 1)/r + f(u) = 0$ and using a shooting method.